In this paper, we study the memory properties of transformations of linear processes. Dittmann and Granger (J Econ 110:113–133, 2002) studied the polynomial transformations of Gaussian FARIMA(0, d, 0) processes by applying the orthonormality of the Hermite polynomials under the measure for the standard normal distribution. Nevertheless, the orthogonality does not hold for transformations of non-Gaussian linear processes. Instead, we use the decomposition developed by Ho and Hsing (Ann Stat 24:992–1024, 1996; Ann Probab 25:1636–1669, 1997) to study the memory properties of nonlinear transformations of linear processes, which include the FARIMA(p, d, q) processes, and obtain consistent results as in the Gaussian case. In particular, for stationary processes, the transformations of short-memory time series still have short-memory and the transformation of long-memory time series may have different weaker memory parameters which depend on the power rank of the transformation. On the other hand, the memory properties of transformations of non-stationary time series may not depend on the power ranks of the transformations. This study has application in econometrics and financial data analysis when the time series observations have non-Gaussian heavy tails. As an example, the memory properties of call option processes at different strike prices are discussed in details.
Heavy tail Long memory Linear process Nonlinear transformation Non-stationary Short memory
Mathematics Subject Classification
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The authors thank the referees and the editors for their careful reading of the manuscript and the valuable comments and suggestions. The authors also thank Micah B. Milinovich for the suggestion on hypergeometric series in the proof of Theorem 2.4.
Bailey WN (1935) General hypergeometric series. Cambridge Tracts in Mathematics and Mathematical PhysicsGoogle Scholar
Hurvich CM, Dro R, Brodsky J (1998) The mean squared error of Geweke and Porter-Hudak’s estimator of the memory parameter of a long-memory time-series. J Time Ser Anal 19:19–46MathSciNetCrossRefMATHGoogle Scholar